Uniformly symmetrizable \(3\times 3\) matrices
type: Item
instance of: scholarly article
title: Uniformly symmetrizable \(3\times 3\) matrices
zbMATH Open document ID: 1056.15012
author: Lorenzo Mencherini
author: Sergio Spagnolo
published in: Linear Algebra and its Applications
publication date: 2004-08-06T00:00:00Z
review text: The Cauchy problem for a hyperbolic system with constant coefficients of the form \(u_t= \sum^n_{j=1} A_j u_{x_j}+ Bu\) is well-posed in \(C^\infty\) for any \(B\), if and only if the family of matrices \(\Sigma_j A_j\xi_j\) is uniformly symmetrizable for \(\xi_1,\dots,\xi_n\) running in the real line \(\mathbb{R}\). \textit{H.-O. Kreiss} [Math. Scand. 7, 71--80 (1959; Zbl 0090.09801)] gave the following characterization of the uniform symmetrizability (US): \[ \{A(t)\}\text{ is US } \Leftrightarrow\|(A(t)- zI)^{-1}\|\leq {C\over\text{Im\,}z}\text{ for }\text{Im\,}z> 0. \] The purpose of this paper is to prove a more intrinsic characterization, at least for matrices of order 2 or 3. Section 2 contains five prelimary remarks. Sections 3 and 4 are proofs of necessary and sufficient conditions for the uniform symmetrizability of a family of matrices. Sections 5 and 6 are for the cases of triangular and \(2\times 2\) matrices.
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 35L45
zbMATH DE Number: 2081900
zbMATH Keywords: symmetrizable matrices
zbMATH Keywords: matrices depending on parameters
zbMATH Keywords: hyperbolic systems
zbMATH Keywords: Cauchy problem
zbMATH Keywords: uniform symmetrizability
reviewed by: Yueh-er Kuo
MaRDI profile type: Publication
full work available at URL: j.laa.2003.11.032
OpenAlex ID: W1986284990
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cites work: Sur le système hyperbolique à coefficients constants
cites work: Über Matrizen die beschränkte Halbgruppen erzeugen.
cites work: Smoothly symmetrizable systems and the reduced dimensions
cites work: On strong hyperbolicity
DOI: 10.1016/J.LAA.2003.11.032
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description: scientific article; zbMATH DE number 2081900